NO-ABEL 1996 Problem 1

Let $S$ be a circle with center $C$ and radius $r$ , and let $P \ne C$ be an arbitrary point. A line $\ell$ through $P$ intersects the circle in $X$ and $Y$ . Let $Z$ be the midpoint of $XY$ . Prove that the points $Z$ , as $\ell$ varies, describe a circle. Find the center and radius of this circle.